Translation secon ODE

Every second order linear ODE a(x)y''+b(x)y'+c(x)y=F(x) can be transformed into the forms;

u''+g(x)u=h(x)

divide whole equation to a(x) and choose y(x)=u(x)^(e^s(x)) but how i choose s(x) ?Please help...
 
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If you mean [itex]y=ue^{s(x)}[/itex], then that's a standard transformation to it's normal form. Write:

[tex]y''+Py'+Qy=g[/tex]

then let:

[tex]y=ue^{s(x)}[/tex]

with:

[tex]s(x)=exp\left\{-1/2\int P dx\right\}[/tex]

See "Intermediate Differential Equations" by Rainville
 

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